Essays on treatment effects and moment inequalities
This dissertation examines methods for evaluating the causal effect of a treatment on a binary outcome using observational data, in which the treatment has not been assigned randomly. For example, consider estimating the average effect of attending a private high school on the probability of graduation. Although private schools may have higher graduation rates than public schools, perhaps the students who choose to attend private schools are better students initially and would perform well at public schools too. In such a case, the treatment exhibits selection bias, making it more difficult to estimate the causal treatment effect.
One approach to estimating treatment effects in the presence of selection bias is to find an instrumental variable, which is correlated with the treatment but has no effect on the outcome except through its effect on the treatment. The first chapter, coauthored with Jishnu Das and Michael Lokshin at the World Bank, presents asymptotic theory and Monte-Carlo simulation results comparing two techniques for estimating treatment effects using an instrument: maximum-likelihood bivariate probit and linear instrumental variables. It compares the mean-square error of the estimators and size and power of tests based on these estimators across a wide range of parameter values. Even though the instrumental variables estimator is usually inconsistent for the average treatment effect, it sometimes outperforms maximum likelihood in terms of mean-square error. The instrumental variables estimator is also more robust to a wider variety of specifications. However, both of these estimation approaches have drawbacks. The bivariate probit model requires strong and often unrealistic assumptions, and the instrumental variables estimator is inconsistent.
The second chapter examines how much can be deduced about the overall average treatment effect from weaker, more realistic assumptions. Such assumptions typically yield upper and lower bounds on the average treatment effect rather than point identification, but the results will be more reliable. A variety of semiparametric assumptions are introduced that exploit variation in other observable variables to produce the bounds. In one model, the outcome is determined by a parametric model such as a probit, but the treatment may be arbitrarily endogenous. Most of the bounding strategies do not require the existence of an instrument, but incorporating an instrument narrows the bounds. The bounds are further improved with the assumption that the treatment and outcome follow a joint threshold-crossing process.
All of the bounding models in the second chapter are examples of moment inequality models. While the second chapter focuses only on identification of the bounds, the third chapter develops powerful new methods for inference on moment inequality models that can be used to obtain more informative confidence intervals for the average treatment effect. The existing literature on testing moment inequalities has mostly focused on finding appropriate critical values for tests based on ad hoc objective functions, but the power of a test depends on the objective function that was used. In contrast, a new test is developed that nearly maximizes weighted average power against the alternative hypothesis under an asymptotic normal approximation. This test is computationally feasible only for low-dimensional problems, but it can be approximated by a simpler test that is feasible in higher dimensions. The acceptance region of the latter test is the union of acceptance regions of likelihood-ratio tests for subsets of the constraints. In most cases the test is exactly the most powerful test within an intuitively reasonable class of tests, and simulations suggest that it outperforms other tests in the literature and comes close to attaining the maximum possible weighted average power.
0511: Economic theory